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Comprehensive Introduction to Linear Algebra
PART II - POLYNOMIALS AND CANONICAL FORMS
Joel G. Broida
S. Gill Williamson
Creative Commons CC0 1.0, 2012
Preface (Part II)
This book, Part 2 - Polynomials and Canonical Forms, covers Chapters 6 through 8 of the book
A Comprehensive Introduction to Linear Algebra(Addison-Wesley, 1986), by JoelG. Broida and
S. Gill Williamson. Chapter 6, Polynomials, will be review to some readers and new to others.
Chapters 7 and 8 supplement and extend ideas developed in Part I, Basic Linear Algebra, and
introduce the very powerful method of canonical forms. The original Preface, Contents and
Index are included.
v
Preface
As a text, this book is intended for upper division undergraduate and begin-
ning graduate students in mathematics, applied mathematics, and fields of
science and engineering that rely heavily on mathematical methods. However,
it has been organized with particular concern for workers in these diverse
fields who want to review the subject of linear algebra. In other words, we
have written a book which we hope will still be referred to long after any final
exam is over. As a result, we have included far more material than can possi-
bly be covered in a single semester or quarter. This accomplishes at least two
things. First, it provides the basis for a wide range of possible courses that can
be tailored to the needs of the student or the desire of the instructor. And
second, it becomes much easier for the student to later learn the basics of
several more advanced topics such as tensors and infinite-dimensional vector
spaces from a point of view coherent with elementary linear algebra. Indeed,
we hope that this text will be quite useful for self-study. Because of this, our
proofs are extremely detailed and should allow the instructor extra time to
work out exercises and provide additional examples if desired.
vii
viii PREFACE
A major concern in writing this book has been to develop a text that
addresses the exceptional diversity of the audience that needs to know some-
thing about the subject of linear algebra. Although seldom explicitly
acknowledged, one of the central difficulties in teaching a linear algebra
course to advanced students is that they have been exposed to the basic back-
ground material from many different sources and points of view. An experi-
enced mathematician will see the essential equivalence of these points of
view, but these same differences seem large and very formidable to the
students. An engineering student for example, can waste an inordinate amount
of time because of some trivial mathematical concept missing from their
background. A mathematics student might have had a concept from a different
point of view and not realize the equivalence of that point of view to the one
currently required. Although such problems can arise in any advanced mathe-
matics course, they seem to be particularly acute in linear algebra.
To address this problem of student diversity, we have written a very self-
contained text by including a large amount of background material necessary
for a more advanced understanding of linear algebra. The most elementary of
this material constitutes Chapter 0, and some basic analysis is presented in
three appendices. In addition, we present a thorough introduction to those
aspects of abstract algebra, including groups, rings, fields and polynomials
over fields, that relate directly to linear algebra. This material includes both
points that may seem “trivial” as well as more advanced background material.
While trivial points can be quickly skipped by the reader who knows them
already, they can cause discouraging delays for some students if omitted. It is
for this reason that we have tried to err on the side of over-explaining
concepts, especially when these concepts appear in slightly altered forms. The
more advanced reader can gloss over these details, but they are there for those
who need them. We hope that more experienced mathematicians will forgive
our repetitive justification of numerous facts throughout the text.
A glance at the Contents shows that we have covered those topics nor-
mally included in any linear algebra text although, as explained above, to a
greater level of detail than other books. Where we differ significantly in con-
tent from most linear algebra texts however, is in our treatment of canonical
forms (Chapter 8), tensors (Chapter 11), and infinite-dimensional vector
spaces (Chapter 12). In particular, our treatment of the Jordan and rational
canonical forms in Chapter 8 is based entirely on invariant factors and the
PREFACE ix
Smith normal form of a matrix. We feel this approach is well worth the effort
required to learn it since the result is, at least conceptually, a constructive
algorithm for computing the Jordan and rational forms of a matrix. However,
later sections of the chapter tie together this approach with the more standard
treatment in terms of cyclic subspaces. Chapter 11 presents the basic formal-
ism of tensors as they are most commonly used by applied mathematicians,
physicists and engineers. While most students first learn this material in a
course on differential geometry, it is clear that virtually all the theory can be
easily presented at this level, and the extension to differentiable manifolds
then becomes only a technical exercise. Since this approach is all that most
scientists ever need, we leave more general treatments to advanced courses on
abstract algebra. Finally, Chapter 12 serves as an introduction to the theory of
infinite-dimensional vector spaces. We felt it is desirable to give the student
some idea of the problems associated with infinite-dimensional spaces and
how they are to be handled. And in addition, physics students and others
studying quantum mechanics should have some understanding of how linear
operators and their adjoints are properly defined in a Hilbert space.
One major topic we have not treated at all is that of numerical methods.
The main reason for this (other than that the book would have become too
unwieldy) is that we feel at this level, the student who needs to know such
techniques usually takes a separate course devoted entirely to the subject of
numerical analysis. However, as a natural supplement to the present text, we
suggest the very readable “Numerical Analysis” by I. Jacques and C. Judd
(Chapman and Hall, 1987).
The problems in this text have been accumulated over 25 years of teaching
the subject of linear algebra. The more of these problems that the students
work the better. Be particularly wary of the attitude that assumes that some of
these problems are “obvious” and need not be written out or precisely articu-
lated. There are many surprises in the problems that will be missed from this
approach! While these exercises are of varying degrees of difficulty, we have
not distinguished any as being particularly difficult. However, the level of dif-
ficulty ranges from routine calculations that everyone reading this book
should be able to complete, to some that will require a fair amount of thought
from most students.
Because of the wide range of backgrounds, interests and goals of both
students and instructors, there is little point in our recommending a particular
x PREFACE
course outline based on this book. We prefer instead to leave it up to each
teacher individually to decide exactly what material should be covered to meet
the needs of the students. While at least portions of the first seven chapters
should be read in order, the remaining chapters are essentially independent of
each other. Those sections that are essentially applications of previous
concepts, or else are not necessary for the rest of the book are denoted by an
asterisk (*).
Now for one last comment on our notation. We use the symbol ˙ to denote
the end of a proof, and ! to denote the end of an example. Sections are labeled
in the format “Chapter.Section,” and exercises are labeled in the format
“Chapter.Section.Exercise.” For example, Exercise 2.3.4 refers to Exercise 4
of Section 2.3, i.e., Section 3 of Chapter 2. Books listed in the bibliography
are referred to by author and copyright date.
